The problem asks to find the other angle and sides of a right triangle $ABC$ where:

• $\angle C = 90^\circ$ (since it's a right triangle),
• $c = 25$ (hypotenuse),
• $b = 3$ (one of the legs).

We need to find:

1. The other leg $a$.
2. The angles $\angle A$ and $\angle B$.

Step 1: Finding side $a$ using the Pythagorean theorem

For a right triangle, the Pythagorean theorem states: $a^2 + b^2 = c^2$

Substitute the values of $b$ and $c$:

$a^2 + 3^2 = 25^2$

$a^2 + 9 = 625$

$a^2 = 625 - 9 = 616$

$a = \sqrt{616} \approx 24.8$

So, the length of side $a$ is approximately $24.8$.

Step 2: Finding angle $\angle A$

Use the sine function for $\angle A$: $\sin A = \frac{b}{c} = \frac{3}{25}$

Now, calculate $A$:

$A = \arcsin\left(\frac{3}{25}\right) \approx 6.89^\circ$

Step 3: Finding angle $\angle B$

Since the sum of angles in a triangle is $180^\circ$, and we already know $\angle C = 90^\circ$ and $\angle A \approx 6.89^\circ$, we can find $\angle B$ as:

$\angle B = 90^\circ - \angle A = 90^\circ - 6.89^\circ = 83.11^\circ$

Final Answer:

• $a \approx 24.8$
• $\angle A \approx 6.89^\circ$
• $\angle B \approx 83.11^\circ$

Would you like further details or have any questions?

Here are some related questions:

1. How does the Pythagorean theorem work for other right triangle problems?
2. What are some other methods to calculate angles in a triangle?
3. How would the problem change if the value of $b$ were different?
4. What is the relationship between sine, cosine, and tangent in a right triangle?
5. Can the Pythagorean theorem be applied in non-right triangles?

Tip: Always verify if you're working with a right triangle to decide whether or not the Pythagorean theorem applies.